6. Resultados Financieros
6.2. Estado de resultados a 1 año
The majority share of the processing is done by the GeoStructure Processing Pipeline, and this constitutes the main contribution of this monograph. GeoStructure is a collection of algo- rithms and functions that process the point cloud through various stages and extract the fracture orientation measurements.
Figure 4.5:Outline of functions handled explicitly by the GeoStructure.m pipeline
The pipeline concept is based on the work of several researchers in geology, software, and, image processing and computer vision [24, 48, 23, 63, 52, 59, 3]. In particular, Gigli et al. [24] developed a software package to obtain fracture orientations from Lidar point clouds, which served as a conceptual template, although none of the actual functions or the development process is shared. GeoStructure is run after the PSR (using the Meshlab application) and the stages the software algorithm are as follows:
1. Read point cloud file,
2. Create point cloud volume [point space] of uniform dimension that can be volumetrically searched,
3. Search the volume, one proportionally sized voxel at a time,
4. Evaluate each voxel for the presence of points,
5. If points are present, attempt to fit a best-fit plane to the volume,
6. Segment to the point cloud by grouping planes into macroscale surface regions and dif- ferentiate,
7. Aggregate planes of same region into a singular plane approximating the overall best fitting orientation,
The next section provides a detailed description of the more significant stages of the GeoStruc- ture processing pipeline.
4.7.1
Mahalanobis Noise Mitigation
One of the draw backs of the PSR tessellation method is the creation of erroneous extra tessel- lation, where the detail of the point cloud ends. These can either connect back on themselves, forming a bulbous undefined area, or flatten and extend out to to the maximum possible area of the reconstruction.
Introducing Anisotropic Noise
The PSR will join disconnected ends of the structure together and the mechanism by which this occurs can lead to the bulbous mesh structures(pictured in figure 4.4), characterised by larger tessellations, where the points become sparse. When reduced to a point cloud, the vertices of these larger triangles are rendered as a generally anisotropic-type of sparse noise. This is illustrated here in (a) - (d). (e) shows a wireframe representation of this tessellation example, where every vertex in the bulbous areas of non-structure, results in an unwanted point. This will have a general effect very similar to anisotropic noise. (f) illustrates such an effect.
Work by Brophy [5] into the extraction of structure from point clouds with high density noise, demonstrated the effectiveness of a Mahalanobis distance metric for removing extraneous points.
The Mahalanobis distance operates in the following way:
d
Σ(x,x
i)
=
((x−
x
|i)H
−1(x−
x
i))
1/2,
(4.1)where the covariance matrix
H
=
DD
|,
(4.2)is constructed using the following:
D
=
(x
1−
x,x
2−
x, . . . ,x
n−
x).
(4.3)As a measure of the distance between a point and the distribution of the set along each principle component axis, it is very effective, and, has the properties of being scale invariant, unitless and respecting correlations in the set [5, 64, 52].
(a)σ=2 (b)σ=4 (c)σ=5
(d)σ=2 (e)σ=4 (f)σ=5
Figure 4.6: The above reconstructions compare the effect of a standard Euclidian distance method against a Mahalanobis method. The Euclidean method (a, b, c) shows significant noise with some legitimate structure no longer present. The Mahalanobis method (d, e, f) demonstrates significantly better behaviour at allσdistances.
Understanding the Effectiveness of the Metric
In a comparison of the removal of noisy points using a standard least squares method against using the Mahalanobis distance method, the Mahalanobis distance ratio of signal points re- moved to noise points is much better. With a least squares method of filtering, as the higher distance values are used, voids start to appear in the main structure, because this standard method has the effect of removingallpoints a certain distance from the centroid, which leads to problems where a non uniform structure is the subject.
The Mahalanobis metric recognises individual areas of density and correlation, and recognises those as part of the overall structure. The effect is that the metric is far more likely to recognise particular points as part of a structure (like a wall, for instance) and not remove those, and so, handles much higher distance values while maintaining the structures integrity.
Fig 4.6 illustrates this by showing how the PSR tessellation process can leave noise artefacts in the resulting reconstruction. The above reconstructions compare the effect of a standard Euclidian distance method (a, b, c) against a vanilla Mahalanobis method (d, e, f). The Eu- clidean method shows large numbers of noise clusters at a distance of 4 & 5 standard deviations (
σ
) from the mean, while atσ
=2, many noise clusters still occur, with voids present, over-all structural point density visibly down and some legitimate structure no longer present. In contrast, the Mahalanobis distance demonstrates significantly better behaviour at all
σ
values. Noise clusters are greatly reduced atσ
= 4 & 5, although some minor structural erosion has occurred. Atσ
=2 there is more significant erosion but all noise has been filtered.As mentioned above, this step is important because errant points will generate errant planes, and these with skew the orientation of region planes, later on in the process.